Q2. What are the future works mentioned in the paper "Treewidth: computational experiments" ?
Therefore, this paper should be seen as one of the rst ( together with [ 8 ] ) in a series of the authors and other collaborators to study the practical setting of treewidth and the tractability of the tree decomposition approach for combinatorial optimization problems.
Q3. What is the rst class of graphs for which perfectness has been proved?
Triangulated graphs are among the rst classes of graphs for which perfectness has been be proved (a graph is perfect if its clique number equals its chromatic number).
Q4. What is the easiest way to recognize triangulated graphs?
The easiest way to recognize triangulated graphs is the construction of an elimination scheme , which is perfect (i.e., no edges are added by Algorithm 1) in case the graph is indeed triangulated.
Q5. How long did it take to compute the MSVS bound?
On average it took more than 20,000 seconds (> 5:5 hours) to compute the MSVS bound, whereas the MCS, LEX P, and LEX M bound are computed in respectively 1,768, 1,974, and 2,167 seconds (30{36 minutes).
Q6. What are the heuristics and lower bounds?
Motivated by problems from combinatorial optimization and probabilistic networks, the authors describe a computational analysis of four heuristics and two lower bounds in this paper.
Q7. What is the procedure to solve an optimization problem with for instance bounded treewidth?
The procedure to solve an optimization problem with for instance bounded treewidth involves two steps: (i) computation of a (good) tree decomposition, and (ii) application of an algorithm that solves instances of bounded treewidth in polynomial time (typically by a dynamic programming algorithm based on the tree decomposition).
Q8. What is the ecient algorithm for the inference calculation in probabilistic (?
The currently most eÆcient algorithm for the inference calculation in probabilistic (or Bayesian) networks builds upon a tree decomposition of the network's moralized graph [20, 24].
Q9. what is the st-separating set of a graph?
An st-separating set of a graph G = (V;E) is a set S V n fs; tg with the property that any path from s to t passes through a vertex of S.
Q10. What is the cardinality of the associated vertex sets?
The cardinality of the associated vertex sets decreases, maxfjXi j;maxp2I jXpjg < jXij, if and only if H is not a complete graph.
Q11. What is the requirement for a tree to be replaced by a new node?
For each replacement of a node by new nodes, the new nodes have to be connected with the remaining parts of the tree in such a way that again all conditions of a tree decomposition are satis ed.
Q12. How can the authors compute the maximum clique number?
several studies have shown that for not too large and not too dense graphs, the maximum clique number can be computed within reasonable time [4, 27].
Q13. What is the minimum number of vertex-disjoint paths in a graph?
Given a graph G = (V;E) and two distinct non-adjacent vertices s; t 2 V , the minimum number of vertices in an st-separating set is equal to the maximum number of vertex-disjoint paths connecting s and t.
Q14. What is the requirement for the vertex set Xi of node i?
The authors require that the vertex set Xi of node i de nes a minimum separating vertex set in a graph H = (V (H); E(H)) to be speci ed in the following.
Q15. What is the minimum separating vertex set?
If H is complete, then no minimum separating vertex set exists, and thus no reduction in the cardinality of the associated vertex sets can be achieved.
Q16. How many instances are available for the heuristics?
After the applicationof pre-processing techniques for computing the treewidth [8], an additional four instances to conduct their heuristics on are available.
Q17. What is the alternative to the MSVS bound?
A good alternative to this time consuming bound is the LEX M bound that performs almost as good as MSVS but has the advantage to be computable in substantially less time.